1.1 Many-Electron Wave Functions
7
the result has to be multiplied with 1/
√
N ! to yield a normalized wave function (see
Exercise 1.4).
Owing to the formal equivalence to the definition of matrix determinants, an
antisymmetric product state wave function can also be written in the form of a
determinant of spin-orbitals, referred to as Slater determinant:
A (ξ 1 , . . . , ξ N ) =
1
√
N !
ψ q 1 (ξ 1 ) ψ q 1 (ξ 2 ) · · · ψ q 1 (ξ N )
ψ q 2 (ξ 1 ) ψ q 2 (ξ 2 ) · · · ψ q 2 (ξ N )
. . .
. . .
. . .
. . .
ψ q N (ξ 1 ) ψ q N (ξ 2 ) · · · ψ q N (ξ N )
(1.25)
A shorthand notation for the Slater determinant (including the normalization factor)
is as follows:
A (ξ 1 , . . . , ξ N ) =
ψ q 1 (ξ 1 )ψ q 2 (ξ 2 ) · · · ψ q N (ξ N )
(1.26)
The antisymmetric product states fulfill the following properties:
1. Symmetry with respect to permutations:
ˆ
P|q 1 . . . q N = (−1)
P
|q 1 . . . q N
(1.27)
2. Pauli principle (the wave function vanishes if two electrons occupy the same
one-particle state):
|q 1 . . . q N ≡ 0
if q i = q j , i = j
(1.28)
3. Linear combination of spin-orbitals:
|q 1 . . . q i−1 (aq + bq
)q i+1 . . . q N
= a|q 1 . . . q i−1 qq i+1 . . . q N + b|q 1 . . . q i−1 q
q i+1 . . . q N (1.29)
The (multi-)linearity of the Slater determinants can be generalized to an arbitrary
linear transformation
| ˜
q i =
N
j=1
|q j U ji ,
U ji ∈ C
(1.30)
of the set of spin-orbitals, yielding the expression
| ˜
q 1 . . . ˜
q N = det(U)|q 1 . . . q N
(1.31)
for the Slater determinant of the transformed spin-orbitals. Here, det(U) is the determinant of the matrix of elements U kl . Equation (1.31) can readily be derived (Exercise 1.3) using the properties (1.27)–(1.29).
7
the result has to be multiplied with 1/
√
N ! to yield a normalized wave function (see
Exercise 1.4).
Owing to the formal equivalence to the definition of matrix determinants, an
antisymmetric product state wave function can also be written in the form of a
determinant of spin-orbitals, referred to as Slater determinant:
A (ξ 1 , . . . , ξ N ) =
1
√
N !
ψ q 1 (ξ 1 ) ψ q 1 (ξ 2 ) · · · ψ q 1 (ξ N )
ψ q 2 (ξ 1 ) ψ q 2 (ξ 2 ) · · · ψ q 2 (ξ N )
. . .
. . .
. . .
. . .
ψ q N (ξ 1 ) ψ q N (ξ 2 ) · · · ψ q N (ξ N )
(1.25)
A shorthand notation for the Slater determinant (including the normalization factor)
is as follows:
A (ξ 1 , . . . , ξ N ) =
ψ q 1 (ξ 1 )ψ q 2 (ξ 2 ) · · · ψ q N (ξ N )
(1.26)
The antisymmetric product states fulfill the following properties:
1. Symmetry with respect to permutations:
ˆ
P|q 1 . . . q N = (−1)
P
|q 1 . . . q N
(1.27)
2. Pauli principle (the wave function vanishes if two electrons occupy the same
one-particle state):
|q 1 . . . q N ≡ 0
if q i = q j , i = j
(1.28)
3. Linear combination of spin-orbitals:
|q 1 . . . q i−1 (aq + bq
)q i+1 . . . q N
= a|q 1 . . . q i−1 qq i+1 . . . q N + b|q 1 . . . q i−1 q
q i+1 . . . q N (1.29)
The (multi-)linearity of the Slater determinants can be generalized to an arbitrary
linear transformation
| ˜
q i =
N
j=1
|q j U ji ,
U ji ∈ C
(1.30)
of the set of spin-orbitals, yielding the expression
| ˜
q 1 . . . ˜
q N = det(U)|q 1 . . . q N
(1.31)
for the Slater determinant of the transformed spin-orbitals. Here, det(U) is the determinant of the matrix of elements U kl . Equation (1.31) can readily be derived (Exercise 1.3) using the properties (1.27)–(1.29).